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6 Correlation Function of Fields
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sin Os (3.5.75a)
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+ (lfhh(S, sOb)1 2 )]}
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VB (so)
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(3.5.75d)
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6 Correlation Function of Fields
Because random medium statistics are described by correlation functions, it will be useful to study the correlation of electromagnetic fields. The correlation of electromagnetic fields can often be related to the correlation properties of the random medium or the random rough surface.
3 FUNDAMENTALS OF RANDOM SCATTERING
We have previously studied the third and fourth Stokes parameters, which represent the correlation between the vertical and horizontal polarizations. In this section, we discuss other useful field correlation functions for random scattering.
Angular Correlation Function
The angular correlation function (ACF) describes the correlation of scattered fields in two different directions. Consider a scattered field in direction ks1 with an incident wave in direction kilo The scattered field is denoted by 'l/J(k sl , kil)' Consider another scattered field 'l/J(k s2 , ki2 ) in direction ks2 due to incident wave in direction ki2. Then the angular correlation function is described by
(3.6.1)
It will be shown in later chapters that if the medium is statistically translational invariant in the horizontal direction - for example, with random permittivity E(X, y, z) such that
(E(XI, YI, ZI)E(X2, Y2, Z2))
= 8(XI
- x2)8(YI - Y2)R(XI, YI; Zl, Z2)
(3.6.2)
where R is some correlation function of the random medium ACF is zero unless
then the
(3.6.3)
This can be regarded as a phase matching condition for the angular correlation function for statistical inhomogeneous medium. It is also called the "memory effect."
Mutual Coherent Function
Consider an incident wave at a carrier frequency
Wo .
In general the field
u(r, t) is u(r, t) = Re['l/J(r, t)e- iwot ]
where 'l/J(r, t) is the complex field.
(3.6.4)
'l/J = 'l/Jc + 'l/Jf when 'l/Jc = ('l/J)
The correlation function time points:
(3.6.5)
is the correlation of the fields at two space and
(3.6.6)
Correlation Function of Fields
It can be decomposed into a coherent part and an incoherent part.
f = f c + fi fcCfl, tl;r2, t2) = ('l/J(TI, tl)) ('l/J*(T2, t2)) fi(rl, tl;r2, t2) = ('l/J/(rl, t l )) ('l/Jj(r2, t2)) (3.6.7)
(3.6.8) (3.6.9)
Consider the case that TI = r2 = T and the random medium is stationary in time. Then f i depends only on the time difference
f - f (r , T)
(3.6.10)
where T = tl - t2 is the time difference. The temporal frequency spectrum is the Fourier transform of the correlation function
Wi(r, w) =
dTfi(r, T)e
(3.6.11)
(3.6.12)
Two Frequency Mutual Coherent Function
The concept of correlation function can be generalized to two frequencies. Let u(w, T, t) be the field where the incident wave is a time harmonic function of e-iwt . Then u(w, T, t) = Re('lj!(w, T, t)e- iwt ) (3.6.13) The two-frequency mutual coherent function is
r(WI,TI,tl;w2,T2,t2) = ('lj!(WI,TI,tI)'lj!*(W2,T2,t2))
(3.6.14)
It can also be decomposed into a coherent part and an incoherent part
r c = ('lj!(WI, TI, tl)) ('lj!*(W2, T2, t2))
f=fc+f i
Relation Between Correlation Function and Specific Intensity
(3.6.15)
(3.6.16)
We can write the fluctuating field 'l/J/(r) as consisting of a spectrum of waves going in all 41f directions
'l/J/(rl) = 'lj!j(T2)
dS{;/(TI,s)eiks'Tl
d'S;j; j(r2, s)e-iks.r2
(3.6.17)
(3.6.18)
3 FUNDAMENTALS OF RANDOM SCATTERING
Then
r /('r1, r2)
= ('l/J/(r1)'l/Jj(r2))
dS l
ds 2({;/(r1, Sl){;j(r2, 82))eiksl'Tle-iks2'T2
(3.6.19)
For fluctuating fields, it is usually the case that
({J/("r1, 81){Jj(r2' 82)) = I(r a, 81)02(81 - 82)
(3.6.20)
where I is the specific intensity, 02 is the two-dimensional angular Dirac delta function, and r a = (r1 + r2)/2. Equation (3.6.20) means that the correlation function in spectral domain is uncorrelated for two different directions and the correlation is also zero if r1 and r2 are far apart. Putting (3.6.20) in (3.6.19) gives
r!Cr1,r2) =
dSI(ra,8)eiks'(TI-T2)
(3.6.21)
showing that the field correlation function is an angular transform of the specific intensity.
REFERENCES
REFERENCES AND ADDITIONAL READINGS
Beran, M. J. (1968), Statistical Continuum Theories, Wiley-Interscience, New York. Booker, H. G. and W. E. Gordon (1950), Outline of a theory of radio scatterings in the troposphere, J. Geophys. Res., 55, 241-246. Callen, H. B. and T. A. Welton (1951), Irreversibility and generalized noise, Phys. Rev., 83, 34. Chandrasekhar, S. (1960), Radiative Transfer, Dover, New York. Chow, P. 1., W. E. Kohler, and G. C. Papanicolaou, Eds. (1981), Multiple Scattering and Waves in Random Media, North-Holland Publishing Company, Amsterdam. Foldy, L. L. (1945), The multiple scattering of waves, Phys. Rev., 67, 107-119. Frisch, V. (1968), Wave propagation in random medium, Probabilistic Methods in Applied Mathematics, 1, Bharuch-Reid, Ed., Academic Press. Gasiewski, A. J. and D. B. Kunkee (1994), Polarized microwave emission from water waves, Radio Sci., 29, 1449-1465. Ishimaru, A. (1978), Wave Propagation and Scattering in Random Media, 1 and 2, Academic Press, New York. Johnson, J. T., J. A. Kong, R. T. Shin, S. H. Yueh, S. V. Nghiem, and R. Kwok (1994), Polarimetric thermal emission from rough ocean surfaces, J. Electromag. Waves and Appl., 8, 43-59. Keller, J. B. (1964), Stochastic equations and wave propagation in random media, Proc. Symp. Appl. Math., 16, 145-170 (Am Math. Soc., Providence, RI). Landau, 1. and E. Lifshitz (1960), Electrodynamics of Continuous Media, Pergamon Press, Oxford. Lax, M. (1951), Multiple scattering of waves, Rev. Modern Phys., 23, 287-310. Lax, M. (1952), Multiple scattering of waves II. The effective field in dense systems, Phys. Rev., 85, 261-269. Li, 1., C. H. Chan, and L. Tsang (1994), Numerical simulation of conical diffraction of tapered electromagnetic waves from random rough surfaces and applications to passive remote sensing, Radio Sci., 29(3), 587-598. Marshall, J. S. and J. W. M. Palmer (1948), The distribution of rain drops with size, J. Meteorol., 5, 165-166. Peake, W. H. (1959), Interaction of electromagnetic waves with some natural surfaces, IEEE Tmns. Antennas Propagat., 7, Special Supplement, S324-S329. Ray, P. S. (1972), Broadband complex refractive indices of ice and water, Appl. Optics, 11, 1836-1844. Sarabandi, K. (1992), Derivation of phase statistics of dIstributed targets from the Mueller matrix, Radio Sci., 27(5), 553-560. Tatarskii, V. I. (1971), The effects of turbulent atmosphere on wave propagation, National Tech. Information Service, 472, Springfield, VA. Tsang, 1. (1984), Thermal emission of nonspherical particles, Radio Sci., 19(4), 966-974. Tsang, L. (1991), Polarimetric passive microwave remote sensing of random discrete scatterers and rough surfaces, J. Electromag. Waves and Appl., 5(1), 41-57. Tsang, 1. and Z. Chen (1990), Polarimetric passive and active remote sensing: Theoretical modeling of random discrete scatterers and rough surfaces, Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IGARSS'90), 2201-2203.